•
•
ON NORMALITY VIA CONDITIONAL NORMALITY
by
M. Ahsanullah and Bikas K. Sinha
Institute of Statistics Mimeograph Series No. 1687
June 1986
•
NORTH CAROLINA STATE UNIVERSITY
Raleigh, North Carolina
ON NORMALITY VIA CONDITIONAL NORMALITY
M. Ahsanullah
Rider College
Lawrenceville, NJ
08648-3099
Bikas K. Sinha *
North Carolina State University
Raleigh" NC 27695-8203
ABSTRACT
Consider X
= (X o'
X1 ... , Xp )' and suppose XOIXi = x i (i=1,2, ... ,p)
p
2
N(ao + E1 aix i , a ). Assume further that X.'s (i=0,1, ... ,p) are marginally
identically distributed. Does this imply n9rmality of X? Ahsanullah (Metrika
,
N
(1985), 32, 215-218) raised this question and resolved it in the affirmative
for p=1.
This is, of course, not true for p > 1.
to that effect.
We give a counter-example
Next we prove that exchangeability of the components X ' ... ,
O
X of X along with conditional normality of X (as stated above) indeed ensure
p
o
normality of X.
AMS 1980 Subject Classification:
Key Words and Phrases:
*
Primary 62E10.
Secondary 62J05.
Exchangeable distributions, normal distribution
On leave from Indian Statistical Institute (Calcutta), India.
· Page 2
In this note we address a problem considered recently by Ahsanullah (1985)
regarding characterization of normality via conditional normality (i.e.,
normality in a conditional distribution).
Consider the following statements
regarding the joint distribution of two random variables (X,Y):
Conditionally given Y=y, X _ N(~ + ~y, a 2 ).
~
52
The marginals are identical (up to a change of location and scale).
53
(X,Y) have an exchangeable distribution (up to a change of location
and scale).
Ahsanullah (1985) essentially demonstrated that
joint normality of X and Y.
~
and 52 together imply
It is clear that 53 implies 52.
A multivariate generalization of this result is developed below.
Consider
the following statements regarding the distribution of a (p+1)-component random
vector X = (X o' X1 ' ... , Xp)1 (assumed to possess a non-singular distribution):
P1
P2
2
Conditionally given Xi = x i (i=1,2, ... ,p), Xo - N(~O + ~ixi' aO)·
The marginals are all identical ·(up to a change of location and
scale).
P3
The components of X have an exchangeable distribution (up to a change
of location and scale).
P4
The marginals of X and X are identical (up to a change of location
1
o
and scale).
P5
Conditionally given Xo = xo' X1 = x 1 ' ... , Xk = x k '
k
2
Xk+1 - N(~+1 + Eo ~+1,i xi' a k+1 ) simultaneously for k=0,1,2, ... ,p.
Ahsanullah (1985) conjectured that just f1 and P2 together imply normality of
X.
This is, however, not true.
5ee counter-example at the end.
Using the
result for two variables, one can easily verify that P4 and P5 together imply
normality of X.
X.
We show below that f1 and P3 together also imply normality of
At this moment, this last assertion seems to be rather uninteresting.
(The
Page 3
proof is quite straightforward as well.)
be strengthened.
However, we do not see how it could
See remark at the end.
It is clear that whenever the components of X have an exchangeable
distribution for themselves,
E(X)
= (e, e, ... , e)1 and
D(X)
= 02[(1-p)I + pJ] which
is the well-known intra-class covariance structure.
no loss of generality.
Note that -lip < P < 1.
We take e=o and 0=1
wi~h
According to fl, we have then
that
2
2
where a = p/{l+(p-l)p} and 1 = 00 + a p{l+(p-l)p}.
This means that the joint
density of Xo ' Xl' ... , Xp can be represented as
1
(1)
f(x O'x 1 ,···,Xp ) = constanteexp{- ---2
2°0
where, by hypothesis, f(e) is exchangeable in xo ' xl'
consequent to (1), g(X 1 , ... ,xp ) ;s
likewi~e
••• I
xp and hence,
exchangeable in xl' x 2 '
.. .,
x •
p
1
Set g (xl' ... , xp ) = constanteexp{- ---2
2°0
(2)
where m(e) is exchangeable in its arguments.
Now we choose 01 and 02 in such a way that
P 2
P
= A EOx,.
+ B E.~.Ex.x.
,.,..J , J
o
Page 4
for some A and B with A > B and A + pB >
This gives 0
~
= p/{1+(p-l)p}.
1
o.
Equating both sides, A = 1 = ~2 + 0
1
= 1 - ~2 and O = _ ~ _ ~2 where
2
Now - lip < P < 1 - -1 <
~
< lip and, hence, 0 1 > 0,
01 - O = 1~ > 0, 0 1 + (p-l)02 = 1 - (p-l)~ - P ~2 > O. Further, A - B = 1+0>0,
2
pe
.
A+pB = 1-~ = 1- l+(p-l)p = (1-p)/{1+(p-l)p} > o. Thus we have observed that f
has a representation
f = (a (p+l)-variate normal density with p.d. intra-class covariance structure).
m(x , x , ••• , x ) •
p
l
2
In view of exchangeability of f, it now follows that m(.) is necessarily a constant.
Thus we have established that Pl and P3 together imply normality of X.
Counter example
Let U be an uniformly distributed rv in (0,1), and
Y. i id N(0,1), i=1,2,3,4,5
,
Suppose
So that
Now <1>x IX
3
Xl = NY 1 +J1-UY 2
X = NY +
3
2
h-u Y2
X3 = '/uY4 +
h-u Y5
X. ~ N(0,1) , i=1,2,3
,
X (t)
l' 2
implying thereby that X31X1, X _ N(0,1) .
2
Page 5
However,
= E E(e
it,<JUV , +
h-u
+ it 2 (NV 3 +
V )
2
h-u
V )
2 IU)
-t 2U/2 - t 2U/2 - (t,+t )2(,-U)/2
2
= E(e'
2
2
=e
=
e
-(t,+t 2 ) /2
t,t 2
-,
t,t 2U
E(e)
2
e
-(t,+t 2 ) /2
t,t 2
Thus
x" x2 "7'" BVN
Remark:
and hence
x" x2 , x3 ,1 MVN
.
The above counter-example also shows that exchangeability in the joint
marginal distribution of (X"X 2 ) is not helpful again in settling the normality
REFERENCE
Ahsanullah, M. ('985).
distribution.
Some characterizat,ons of the bivariate normal
Hetrika, 32, 2'5-2'8.